For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means that if $V$ is a finitely generated $G$-rep, then all its submodules are also finitely generated.
If $G$ is a topological group with an interesting topology, then $k[G]$ is almost never what you actually want to look at, since it ignores the topology on $G$. Indeed, what we care about are continuous representations of $G$, and there are various continuous group algebras in this case - $C(G)$, $L^{1,2,∞}(G)$, the algebra of Radon measure $M(G)$, and many variants - with multiplication again given by convolution.
Here the analogous property should be "topologically Noetherian". We want a topologically Noetherian module to be one all of whose closed submodules are topologically finitely generated, and a topologically Noetherian ring to be one all of whose t.f.g. modules are top. Noeth. I'm pretty sure this reduces to the usual property of ACC on closed ideals, but if there's any subtlety, this seems like the right definition.
So, are any of the various group algebras (absolutely including any not mentioned above) topologically Noetherian in this sense for nice $G$, say for $G$ a compact Lie group? My sense is that this is probably too much to hope for.
